Phase retrieval of bandlimited functions for the wavelet transform

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

We investigate the recovery of square-integrable signals from the absolute values of their wavelet transforms, referred to as wavelet phase retrieval. We introduce a novel uniqueness result for wavelet phase retrieval, demonstrating that any wavelet with a finite number of vanishing moments can uniquely recover real-valued bandlimited signals, albeit without determining the exact global sign. Furthermore, we present the first uniqueness result for sampled wavelet phase retrieval using complex-valued wavelets, and a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.